Three Query Locally Decodable Codes with Higher Correctness Require Exponential Length

Locally decodable codes are error correcting codes with the extra property that, in order to retrieve the correct value of just one position of the input with high probability, it is sufficient to read a small number of positions of the corresponding, possibly corrupted codeword. A breakthrough result by Yekhanin showed that 3-query linear locally decodable codes may have subexponential length. The construction of Yekhanin, and the three query constructions that followed, achieve correctness only up to a certain limit which is 1 − 3δ for nonbinary codes, where an adversary is allowed to corrupt up to δ fraction of the codeword. The largest correctness for a subexponential length 3-query binary code is achieved in a construction by Woodruff, and it is below 1 − 3δ. We show that achieving slightly larger correctness (as a function of δ) requires exponential codeword length for 3-query codes. Previously, there were no larger than quadratic lower bounds known for locally decodabl...

• Type: Journal
• Year: 2011
• Where: ECCC
• Authors: Anna Gál, Andrew Mills
• Keywords: Decodable Codes, ECCC 2011, ECommerce, Locally Decodable, Lower Bounds